What Is The Simplified Form Of The Expression 2(⁴√16x) - 2(⁴√2y) + 3(⁴√81x) - 4(⁴√32y) Assuming X ≥ 0 And Y ≥ 0?

When dealing with mathematical expressions involving radicals, simplification is a crucial step to arrive at the most concise and understandable form. This process often involves identifying perfect powers within the radicand (the expression under the radical) and extracting them. In this comprehensive guide, we will delve into the simplification of a specific expression containing fourth roots, assuming non-negative values for the variables involved. We'll break down each step, providing a clear and detailed explanation to enhance understanding.

The Expression: Let's consider the expression: 2(⁴√16x) - 2(⁴√2y) + 3(⁴√81x) - 4(⁴√32y). This expression involves fourth roots of terms containing variables 'x' and 'y'. Our goal is to simplify this expression, assuming that both 'x' and 'y' are greater than or equal to zero. This assumption is essential because the fourth root of a negative number is not a real number. The simplification process will involve identifying perfect fourth powers within the radicands and extracting them from the radical. We will also combine like terms, which are terms that have the same radical part. This step-by-step approach will help us to arrive at the simplified form of the expression.

Breaking Down the Radicals: The first step in simplifying the expression is to break down each radical term individually. This involves identifying the largest perfect fourth power that divides the radicand. For example, in the term ⁴√16x, we recognize that 16 is a perfect fourth power (2⁴ = 16). Similarly, in ⁴√81x, 81 is a perfect fourth power (3⁴ = 81). For the terms ⁴√2y and ⁴√32y, we need to further break down the radicands to identify any perfect fourth power factors. By breaking down each radical term, we can simplify the expression more easily. This step is crucial for extracting the perfect fourth powers from the radicals, which is a key part of the simplification process. It also helps in combining like terms later on.

Simplifying Individual Terms: Let's simplify each term separately:

1. 2(⁴√16x): Since 16 is 2⁴, we have 2(⁴√2⁴x) = 2 * 2(⁴√x) = 4(⁴√x). This simplification involves recognizing that 16 is a perfect fourth power and extracting the fourth root of 16, which is 2. The coefficient 2 outside the radical is then multiplied by the extracted 2, resulting in a coefficient of 4. The remaining term under the radical is x, which is left as ⁴√x since we assume x ≥ 0.
2. -2(⁴√2y): The number 2 does not have any perfect fourth power factors other than 1. Therefore, this term remains as -2(⁴√2y). This term cannot be simplified further because 2 does not have any factors that are perfect fourth powers. The term is left as it is, which means it will be combined with other similar terms later in the simplification process.
3. 3(⁴√81x): Since 81 is 3⁴, we have 3(⁴√3⁴x) = 3 * 3(⁴√x) = 9(⁴√x). In this case, 81 is a perfect fourth power, and its fourth root is 3. This 3 is multiplied by the coefficient 3 outside the radical, resulting in a coefficient of 9. The remaining term under the radical is x, which is left as ⁴√x.
4. -4(⁴√32y): We can write 32 as 16 * 2, where 16 is 2⁴. Thus, -4(⁴√32y) = -4(⁴√2⁴ * 2y) = -4 * 2(⁴√2y) = -8(⁴√2y). Here, 32 is factored into 16 * 2, where 16 is a perfect fourth power (2⁴). The fourth root of 16 is 2, which is multiplied by the coefficient -4 outside the radical, resulting in a coefficient of -8. The remaining term under the radical is 2y, which is left as ⁴√2y.

Combining Like Terms: Now, we substitute the simplified terms back into the expression: 4(⁴√x) - 2(⁴√2y) + 9(⁴√x) - 8(⁴√2y). Like terms are those that have the same radical part. In this expression, 4(⁴√x) and 9(⁴√x) are like terms, and -2(⁴√2y) and -8(⁴√2y) are like terms. Combining the like terms, we have: (4 + 9)(⁴√x) + (-2 - 8)(⁴√2y) = 13(⁴√x) - 10(⁴√2y). This step involves adding the coefficients of the like terms while keeping the radical part the same. The result is a simplified expression with two terms.

Final Simplified Expression: The simplified form of the expression is 13(⁴√x) - 10(⁴√2y). This is the most concise form of the original expression, where we have extracted all perfect fourth powers from the radicals and combined like terms. The final simplified expression is a combination of terms involving the fourth root of x and the fourth root of 2y. This expression is easier to work with and understand compared to the original expression.

In conclusion, simplifying radical expressions involves a series of steps including breaking down the radicals, simplifying individual terms by extracting perfect powers, and combining like terms. By following these steps systematically, we can transform complex expressions into simpler, more manageable forms. The simplified form, 13(⁴√x) - 10(⁴√2y), represents the original expression in its most concise and understandable form, making it easier to work with in further mathematical operations or analyses. Mastering the techniques of simplifying radical expressions is essential for success in algebra and other advanced mathematical fields. It allows for a clearer understanding of mathematical relationships and facilitates problem-solving by reducing complexity.

Simplifying radical expressions, fourth roots, perfect fourth powers, radicand, like terms, mathematical expressions, algebraic simplification, extracting radicals, combining terms, concise form.